This chapter takes a closer look at a class of MIMO detention methods, collectively referred to as relaxation detectors. These detectors provide computationally advantageous alternatives to the optimal maximum likelihood detector. Previous analysis of relaxation detectors have mainly focused on the implementation aspects, while resorting to Monte Carlo simulations when it comes to investigating their performance in terms of error probability. The objective of this chapter is to illustrate how the performance of any detector in this class can be readily quantified thought its diversity gain when applied to an i.i.d. Rayleigh fading channel, and to show that the diversity gain is often surprisingly simple to derive based on the geometrical properties of the detector.
A central component of wireless multiple input-multiple output (MIMO) systems is the symbol detector or demodulator where the receiver produces estimates of the symbols (or bits) transmitted over the MIMO channel given a set of received signals and an estimate of the channel state. However, unlike their single input-single output (SISO) equivalents, naive implementations of optimal detectors for MIMO channels often tend to be prohibitively computationally complex. This is partially due to the fact that in MIMO channels, each output signal tends to be influenced by all input signals. The nature of this influence is determined by the channel fading which is not a priori known.
The MIMO channel is frequently modeled in vector/matrix form according to (1)
is the vector of received signals
(after matched filtering and sampling), where
is the vector of transmitted symbols
drawn from some constellation alphabet
is additive noise
, and where
is the channel matrix
modeling the input-output relation of the MIMO channel. In the particular case of the narrow-band multiple antenna channel with spatial multiplexing across antennas, the elements of
would have the physical interpretation of baseband equivalent complex gains between the transmitting and receiving antennas (Tse & Viswanath, 2005). The model in itself is however more general and essentially applicable to any scenario where a group of symbols are linearly modulated and transmitted over a linear channel (Barbosa, 1989). In some cases, the channel matrix will have special structure that is exploitable in the transmission and detection process. However, in the multiple antenna MIMO scenario the matrix,
, is often unstructured and this calls for general detection methods.
Under the assumption of uncorrelated Gaussian noise and that the receiver has access to both as well as the maximum likelihood (ML) detector of can be expressed according to (2)